16
11.1.5 Remark Textbooks on logic dene propositions (and predicates, the subalgebraic expression 16
ject of the next chapter) rather than merely specifying them as we have done. The
instance 16
denition is usually by an recursive process and can be fair
40
biconditional 40
conclusion 36
denition 4
divide 4
equivalence 40
equivalent 40
hypothesis 36
implication 35, 36
predicate 16
rule of inference 24
truth table 22
28. Modus Ponens
The truth table for implication may be summed up by saying:
An implicatio
37
26. Vacuous truth
The last two lines of the truth table for implication mean that if the hypothesis of
an implication is false, the implication is automatically true.
26.1 Denition: vacuously true
In the case that P Q is true because P is false, the im
42
30. Statements related to an implication
contrapositive 42
converse 42
decimal expansion 12
30.1 Denition: converse
decimal 12, 93
The converse of an implication P Q is Q P .
denition 4
equivalent 40
30.1.1 Example The converse of
implication 35, 36
If
43
30.4.3 Example Lets look again at this (true) statement (see Section 10, contrapositive 42
converse 42
page 14):
If the decimal expansion of a real number r has all 0s after a certain
point, it is rational.
The contrapositive of this statement is that
44
denition 4
equivalent 40
hypothesis 36
implication 35, 36
include 43
proof 4
properly included 44
set 25, 32
vacuous 37
31.2 Theorem
a) For any set A, A A.
b) For any set A, A.
c) For any sets A and B , A = B if and only if A B and B A.
Proof Using Den
45
31.4.3 Remark The fact that A A for any set A means that any set is a subset denition 4
of itself. This may not be what you expected the word subset to mean. This leads include 43
nontrivial subset 45
to the following denition:
31.5 Denition: proper
A
46
denition 4
empty set 33
fact 1
include 43
powerset 46
rule of inference 24
setbuilder notation 27
set 25, 32
subset 43
32. The powerset of a set
32.1 Denition: powerset
If A is any set, the set of all subsets of A is called the powerset of A
and is den
47
33. Union and intersection
33.1 Denition: union
For any sets A and B , the union A B of A and B is dened by
A B = cfw_x | xA xB
(33.1)
33.2 Denition: intersection
For any sets A and B , intersection A B is dened by
A B = cfw_x | xA xB
(33.2)
33.2.1 Exa
50
coordinate 49
denition 4
integer 3
ordered pair 49
ordered triple 50
specication 2
tuple 50, 139, 140
union 47
usage 2
35.1.2 Method
To prove two ordered pairs x, y and x , y
x = x and y = y .
are the same, prove that
35.1.3 Exercise Which of these pai
51
36.2.3 Example 1, 3, 3, 2 is a tuple of integers. It has length 4. The integer coordinate 49
empty set 33
3 occurs as an entry in this 4-tuple twice, for i = 2 and i = 3.
36.2.4 Usage Tuples and their coordinates are often named according to a subscrip
53
37.5 Denition: Cartesian product
Let A1 , A2 , . . . , An be sets in other words, let Ai
of sets. Then A1 A2 An is the set
a1 , a2 , . . . , an | (i:n)(ai Ai )
in
be an n-tuple
(37.1)
of all n-tuples whose ith coordinate lies in Ai .
37.5.1 Example The
35
24. Russells Paradox
The setbuilder notation has a bug: for some predicates P (x), the notation
cfw_x | P (x) does not dene a set. An example is the predicate x is a set. In
that case, if cfw_x | x is a set were a set, it would be the set of all sets.
36
antecedent 36
conclusion 36
conditional sentence 36
consequent 36, 121
denition 4
hypothesis 36
implication 35, 36
logical connective 21
material conditional 36
predicate 16
truth table 22
type (of a variable) 17
usage 2
Implications are at the very he
33
21.2.2 Example For x real,
cfw_x | x2 = 1 = cfw_x | (x = 1) (x = 1)
We will prove this using Method 21.2.1. Let
A = cfw_x | x2 = 1 and B = cfw_x | (x = 1) (x = 1)
Suppose x A. Then x2 = 1 by 18.2. Then x2 1 = 0, so (x 1)(x + 1) = 0, so
x = 1 or x = 1.
21
14. Logical Connectives
Predicates can be combined into compound predicates using combining words called
logical connectives. In this section, we consider and, or and not.
14.1 Denition: and
If P and Q are predicates, then P Q (P and Q) is also a predi
25
15.2 Denitions and Theorems give rules of inference
divide 4
What Method 3.1.1 (page 4) says informally can be stated more formally this way: integer 3
natural number 3
Every denition gives a rule of inference.
nonnegative integer 3
Similarly, any Theo
24
and 21, 22
denition 4
logical connective 21
or 21, 22
propositional variable 104
rule of inference 24
usage 2
15. Rules of Inference
15.1 Denition: rule of inference
Let P1 , P2 , . . . Pn and Q be predicates. An expression of the form
P1 , . . . , Pn
23
14.5.2 Fact Negation has the very simple truth table
P
T
F
P
F
T
14.5.3 Usage
a) Other notations for P are P and P .
b) The symbol in Mathematica for not is !, the exclamation point. P is
written !P.
c) The symbol always applies to the rst predicate af
26
denition 4
integer 3
set 25, 32
type (of a variable) 17
16.3 Denition:
If x is a member of the set A, one writes x A; if it is not a member
of A, x A.
/
16.3.1 Example 4 Z, 5 Z, but 4/3 Z.
/
16.4 Sets, types and quantiers
When using the symbol , as in
22
denition 4
even 5
fact 1
integer 3
negation 22
or 21, 22
positive integer 3
predicate 16
truth table 22
usage 2
14.3 Truth tables
The denitions of the symbols and can be summarized in truth tables:
P
T
T
F
F
Q
T
F
T
F
P Q
T
F
F
F
P
T
T
F
F
Q
T
F
T
F
P
27
17.1.4 Exercise How many elements does the set cfw_1, 1, 2, 2, 3, 1 have? (Answer comprehension 27,
29
on page 243.)
17.2 Sets in Mathematica
In Mathematica, an expression such as
cfw_2,2,5,6
denotes a list rather than a set. (Lists are treated in deta
29
18.1.11 Method: Comprehension
Let P (x) be a predicate and let A = cfw_x | P (x). Then if you know that
a A, it is correct to conclude that P (a). Moreover, if P (a), then you
know that a A.
18.1.12 Remark The Method of Comprehension means that the ele
31
19.2.8 Exercise How many elements does the set
cfw_
1
1 1
| x = , , 2, 2
2
x
2 2
have?
19.3 More about sets in Mathematica
The Table notation described in 17.2 can use the variations described in 19. For
example, Table[k2,cfw_k,1,5] returns cfw_1,4,9,1
30
and 21, 22
integer 3
predicate 16
rational 11
real number 12
set 25, 32
unit interval 29
19.1.1 Example The unit interval I could be dened as
I = cfw_x R | 0 x 1
making it clear that it is a set of real numbers rather than, say rational numbers.
19.2 O
32
real number 12
setbuilder notation 27
set 25, 32
specication 2
20.2 Bound and free variables
The variable in setbuilder notation, such as the x in Equation (18.3), is bound, in
the sense that you cannot substitute anything for it. The dummy variable x
52
Cartesian product 52 37. Cartesian Products
coordinate 49
denition 4
37.1 Denition: Cartesian product of two sets
diagonal 52
LetA and B be sets. A B is the set of all ordered pairs whose rst
factor 5
coordinate is an element of A and whose second coor
54
Cartesian powers 54
Cartesian product 52
Cartesian square 54
implication 35, 36
include 43
powerset 46
set 25, 32
singleton 34
tuple 50, 139, 140
union 47
37.7.1 For all sets A, B and C , A C = B C A = B .
244.)
(Answer on page
37.7.2 For all sets A an
57
39.2.1 Warning This specication for function is both complicated and subtle
and has conceptual traps. One of the complications is that the concept of function
given here carries more information with it than what is usually given in calculus
books. One
76
denition 4
empty set 33
equivalent 40
function 56
integer 3
ordered pair 49
powerset 46
relation 73
singleton 34
subset 43
As we have seen, the concept of relation from A to B is a generalization of the
concept of function from A to B . In general, for
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